Question

on a cross-country bicycle trip, participants ride about 75 miles per day. approximately how many days, (d), must they ride to travel at least 4,325 miles?

part a

which inequality represents the situation?

\\(\frac{d}{75} \le 4,325\\)
\\(\frac{d}{75} \ge 4,325\\)
\\(75d \le 4,325\\)
\\(75d \ge 4,325\\)

part b

what does the solution to the inequality represent in this situation?

a participant must ride at least 57 days to travel 4,325 miles.
a participant must ride at least 58 days to travel 4,325 miles.
a participant must ride at least 59 days to travel 4,325 miles.
a participant must ride at least 60 days to travel 4,325 miles.

Explanation:

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<reasoning>

Set up the inequality for Part A

\[

$$\begin{aligned} &\text{Rate} = 75\text{ miles/day}\\ &\text{Total distance in } d \text{ days} = 75d\\ &\text{At least } 4,325\text{ miles} \implies 75d \ge 4,325 \end{aligned}$$

\]

Solve the inequality for Part B

\[

$$\begin{aligned} &75d \ge 4,325\\ &d \ge \frac{4,325}{75}\\ &d \ge 57.67 \end{aligned}$$

\]

Interpret the solution in context

\[

$$\begin{aligned} &d \ge 57.67 \implies d \ge 58 \text{ (since days must be whole numbers to complete the distance)}\\ &\text{A participant must ride at least 58 days to travel 4,325 miles.} \end{aligned}$$

\]
</reasoning>

<answer>

Question 1

<mcq-option>(A) \(\frac{d}{75} \le 4,325\)</mcq-option>
<mcq-option>(B) \(\frac{d}{75} \ge 4,325\)</mcq-option>
<mcq-option>(C) \(75d \le 4,325\)</mcq-option>
<mcq-correct>(D) \(75d \ge 4,325\)</mcq-correct>

Question 2

<mcq-option>(A) A participant must ride at least 57 days to travel 4,325 miles.</mcq-option>
<mcq-correct>(B) A participant must ride at least 58 days to travel 4,325 miles.</mcq-correct>
<mcq-option>(C) A participant must ride at least 59 days to travel 4,325 miles.</mcq-option>
<mcq-option>(D) A participant must ride at least 60 days to travel 4,325 miles.</mcq-option>
</answer>

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Answer:

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<reasoning>

Set up the inequality for Part A

\[

$$\begin{aligned} &\text{Rate} = 75\text{ miles/day}\\ &\text{Total distance in } d \text{ days} = 75d\\ &\text{At least } 4,325\text{ miles} \implies 75d \ge 4,325 \end{aligned}$$

\]

Solve the inequality for Part B

\[

$$\begin{aligned} &75d \ge 4,325\\ &d \ge \frac{4,325}{75}\\ &d \ge 57.67 \end{aligned}$$

\]

Interpret the solution in context

\[

$$\begin{aligned} &d \ge 57.67 \implies d \ge 58 \text{ (since days must be whole numbers to complete the distance)}\\ &\text{A participant must ride at least 58 days to travel 4,325 miles.} \end{aligned}$$

\]
</reasoning>

<answer>

Question 1

<mcq-option>(A) \(\frac{d}{75} \le 4,325\)</mcq-option>
<mcq-option>(B) \(\frac{d}{75} \ge 4,325\)</mcq-option>
<mcq-option>(C) \(75d \le 4,325\)</mcq-option>
<mcq-correct>(D) \(75d \ge 4,325\)</mcq-correct>

Question 2

<mcq-option>(A) A participant must ride at least 57 days to travel 4,325 miles.</mcq-option>
<mcq-correct>(B) A participant must ride at least 58 days to travel 4,325 miles.</mcq-correct>
<mcq-option>(C) A participant must ride at least 59 days to travel 4,325 miles.</mcq-option>
<mcq-option>(D) A participant must ride at least 60 days to travel 4,325 miles.</mcq-option>
</answer>

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